// Numbas version: exam_results_page_options
{"name": "Parametric 2nd derivative", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Parametric 2nd derivative", "tags": [], "metadata": {"description": "Calculate d2y/dx2 for a curve defined parametrically", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>A curve is defined by the parametric equations $x=\\var{a}+\\var{b}t^2, y = \\var{c}t+\\var{d}t^3$</p>", "advice": "<p>We have $x=\\var{a}+\\var{b}t^2, y = \\var{c}t+\\var{d}t^3$</p>\n<p>So&nbsp;$\\frac{dx}{dt} = 2 \\times\\var{b}t = \\var{2b}t$</p>\n<p>and&nbsp;$\\frac{dy}{dt} = \\var{c}+3 \\times\\var{d}t^2 = \\var{c}+\\var{3d}t^2$</p>\n<p>Since&nbsp;$\\frac{dy}{dx}=\\frac{dy}{dt}/\\frac{dx}{dt}$ we have&nbsp;$\\frac{dy}{dx}=\\frac{\\var{c}+\\var{3d}t^2}{\\var{2b}t}=\\frac{\\var{c}}{\\var{2b}t}+\\frac{\\var{3d}t}{\\var{2b}}$</p>\n<p></p>\n<p>To find $\\frac{d^2y}{dx^2}$ we use the chain rule to obtain&nbsp;$\\frac{d^2y}{dx^2}=\\frac{d}{dx}(\\frac{dy}{dx})=\\frac{d}{dt}(\\frac{dy}{dx})\\times\\frac{dt}{dx}=\\frac{\\frac{d}{dt}(\\frac{dy}{dx})}{\\frac{dx}{dt}}$</p>\n<p>Now $\\frac{d}{dt}(\\frac{dy}{dx}) =&nbsp;\\frac{-\\var{c}}{\\var{2b}t^2}+\\frac{\\var{3d}}{\\var{2b}}=\\frac{-\\var{c}+\\var{3d}t^2}{\\var{2b}t^2}$</p>\n<p>Hence $\\frac{d^2y}{dx^2}=\\frac{\\frac{d}{dt}(\\frac{dy}{dx})}{\\frac{dx}{dt}}=\\frac{-\\var{c}+\\var{3d}t^2}{\\var{2b}t^2\\times\\var{2b}t}=\\frac{\\var{3d}t^2-\\var{c}}{\\var{4b^2}t^3}$</p>\n<p></p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2,4,5)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(-4 .. 4#1)", "description": "", "templateType": "randrange"}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "(x-a)/b", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "x", "t"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Find&nbsp;$\\frac{d^2y}{dx^2}$ in terms of $t$&nbsp; &nbsp; [[0]]</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 1", "marks": "3", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({3d}t^2-{c})/({4b^2}t^3)", "answerSimplification": "basic,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "t", "value": ""}]}], "sortAnswers": false}], "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}